Our fingers are called "digits",
and so also are the numerals 0,
1, 2, 3, 4, 5, 6, 7, 8, and 9.
("Numerals" means "number characters".) We use these
ten digits, along with the concept of place value, in exactly the same
sense that we were using our fingers and our piles of marbles: a certain
"place" tells us what unit we're working with, and the digit
tells us how many we need of that unit.

So, for instance, the expression "264"
means "two 100s,
plus six 10s,
plus four 1s",
because hundreds, tens, and ones are what are stored in those particular
places. In "expanded notation", "264"
can be written as:

200 + 60 + 4 = 2×100 + 6×10
+ 4×1

Our place-value system is called the "decimal"
system, because it's based on "all my fingers" being "ten
fingers"; that is to say, the "base" of our number system
is ten. We start with ones, being our fingers (that is, our counting "units").
Then we go to tens, being sets of fingers. Then we go to ten sets of sets
of fingers, which is 10×10
= 100; that is, we go to hundreds.
In other words, every time we go one "place" further to the
left (that is, every time we go into a unit that is one times bigger than
the previous place's unit), we multiply by our base of ten:

To "expand" this number, I
need to split it up into its different places. If I'm not sure of my
places, I'll count them out, starting from the right-hand digit.

This number has five digits. From the
table above (if I haven't memorized this information yet), I know
that this means that I'm dealing with tens of thousands. The one comma
tells me that I'm dealing with thousands, too; one comma means I'm into
the thousands, two commas would have meant I'm into the millions, and
so forth.

So I've got three 10,000s,
two 1,000s,
zero 100s,
six 10s,
and seven 1s.
Usually, we ignore the zeroes in expanded notation, so this gives me:

30,000
+ 2,000 + 60 + 7

Write the standard
form for the number which, in expanded notation, is written as follows:
9,000
+ 300 + 2

I've got nine thousands, three hundreds,
and two ones. I don't have any tens in the expanded form, so I'll need
to use a zero in the tens place to keep that slot open. Then my standard
form is:

9,000 + 300 + 0 + 2 = 9,302

For the number 52,973,
(a) state the place held by the 2,
and (b) state the digit in the tens place.

a) The 2 is
immediately to the left of the comma. I only have the one comma, so
I know this number only goes into the thousands. In the thousands, I've
got "52",
so the 2 is
in the thousands place. (The 5 is
in the ten-thousands place.)

b) The tens place is the second place,
just to the left of the 3 in
the ones place. There is a 7 in
this second place, so the
digit in the tens place is 7.

State the number 622,937,285 in words.

The commas break this number into digestible
pieces. The "285"
is in terms of ones, tens, and hundreds of ones (or of just "regular
numbers", in extremely informal language). The "937"
is in terms of ones, tens, and hundreds of thousands. And the
"622"
is in terms of ones, tens, and hundreds of millions.

millions

thousands

(regular)

622

937

285

So the number they gave me, when I spell
it out in words, is six
hundred twenty-two million, nine hundred thirty-seven thousand, two
hundred eighty-five.